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  1. 1. fhe Mathematics Education SECTION BV ol. VI I I, N o . 2 , J u n e 1 97 * G L I M PS E S OF A N C IE N T IN D IAN tv|ATHEM ATICSNO. 10 fBrahrna$uptas Forrnulas for the Area and Dla$o- nals of a Cycltc Gluadrllateral D7R. C. Gupta Asristant Professor Mathematics of of Birla Institute Techuology P.O. Mesra,RANCHI. (Received l5 APril 1974) I fntroduction: Let ABCD bea plane (convex) quadrilateral with sidesAB, BC,CD and DA equal to a, b, c and d respectively. Let the figure be drawn in such a manner that we may corrsider,according to the traditional terminoiogy, the side BC to be the base (bh[)the side AD to be the face (mukha), and the sides AB and DCto be the flank sides (bhu-jas or arms) of the quadrilateral. Since a quadrilareral is not uniquely defined by its four sides, its shape and sizearc not fixed. So that, by merely specifying the four sides, the question of finding itsarea does not arise. arise. That is why erarrq Aryabhat II ( 950 A. D. ) in his cgrfir€raMahi-siddhanta (-lvIS), XV, 70 saysr. +rrinr*t fqqr sg{d qr.[+ si{ qil I qlss] ${: EEd qrsqfdrrqrs] fq{rs} st lleoll Karna-jirZrreua vintr caturasre lambakar.nphalalr yadva / Vaktu4r vErlchati gairako yosau mfirkhah pidaco va llTOll The mathematician who desires to tell the area or the altitude of a quadrilateralwithout knowing a diagonal, is either a fool or a devil. Brahmagupta ( 628 A. D. )t in his ilr€EiuFsdrd Brahma-sphuta-siddbanta (- BSS) has given two rules (see below) for finding the area of a quadrilateral in terms of itsfourgiven sides. One of the rules is forgetting a rough value of the area and the otherfor an accurate (s[ksma) value. Now, Brahmaguptas forrnula for the accurate are of a qua-drilateral gives the cxact value only when the quadrilateral is cyclic, although he hasnot sptcified this condition, But the condition may be taken to be understood, especially whenwe know (see below) that his expressions for the diagonals of the quadrilateral are alsotrue only whcn the figure is cyclic. otherwise the diagonals have remained undefined. Infact, Brahmagupta docs speak of the circum circle (koqasplg-vltta) and the circum-radius
  2. 2. ,t Ht M AIIHEnAIIICB ED U OAT T oN 3+ which ( hldaya-rajju ) otrrianglc arrd quadrilateral in colule(ri.rn rvitlt some other rules2 are givcrr bltrvr-.err lris rrrle frlr the alea and that fol the diagonals of thc ( cyclic ) qtradrilatcritl . 2 Rules for the Area : l h" BSS, XtI, 2l V, r l. I I , p. Bl6) s t at t s (Ti|$q ft<ggwargxfter6a)rl<<era: I qSqII ll Rt ll glalirrtf ?Euaq gqlqqTilIicE Stlrrlla-phalarlrtricatu r blruja-beltupratibrhu-yogadalaglfi tah / B h rrjayogardha-catus a bhLrjorra-ghatatpada4r tay sIkiamam //2 | // f he pr odu <t o l h a l f (h e s rrrn s l (th e two pai rs of ) the opposi te si desof a tri angl e oor a qrradrilateral, gives the gross area. Set down half the sum of the sides irr f<rrrr pla-ces (and) diminish them by the (four) sides (respectively).The square-root of the prod-d u rt ( of t he f our n u m l te rs ) i s th e a c c u ra tea re a . gross area : * (a * c ) . ! , t ( b- f d) (r) accuratearea : l(-s-4t G4t Gt (r-l) (2) where r: (a+1,+c*d12 The above ftrrmulas are stated to be applicable to the quatlrilateral as well as tothe triangle in which case we have to takr: the face d to be zero. Thus, in the case of a tria-n q l t: r , f s ic lc s b . c, w e h a v e a, gr os sar c a :( b l z ). ( a -l c )l l (4) accrrrateirrea -/ s(s_a) ar_r) tr_r) (s) $ lr t : ie s :( a f-b + c )l ) (6) W e s eet hat i t d o e s n o t ma tte r m rrc h rv h etherthe fol mrrl a (l ) i s uscd for cycl i c oro th e l c lr r adlr llt c la l s ,s i n c e , ; fre r a l l , i t i s s ta te d to be a rough one onl l . Fermul a (2) i skrrown tr give ex rct are.r r,nly in the ca:e of cyclic qu.adrilateral. However, the formula (5)is applicable to erery triangle. Rut thc foln.irrla(4) has now an additional defect of notvi e l d in, ; 21111iqe ( th o u g h ro u g l r ) v a l u e o f th e area ol a tri angl e, because w c may get rrl i fl rr r entr es r il s by rc g a rd i ri g e a c h o f th e s i d e sa, b, c to be base i n turn, A ny way , eq tri v a l e n tru l e s ; w h i c h y i e l d formul as (l ) and (2), have been gi ven bysevelal sultsequer Indian rvriters with or without some additional comments. t Some ofth e s ewill be not e, l n r,u ,. Srilhara ( 4tq< ) in lris qrdtqfua Pfttiganita (:PG)a has reproduced, word by word,BSS rule rvhich gives the formrrla (l). However. he adds the following remarks immedia-tcly aftelv <luotingthe rules; But tl:is result (l) it true only f..r those figures in r,vhichthe difference between thealtitude and the flank sidesis small. In the caseof orher fisures the above result is far
  3. 3. R. ( j. GUPT A :.t5 removed from the truth; as for example, in the caseof the triangle having I3 for the tr.,"o (flarrk) sidesand 24for the base, the gross area is 156, whereas the correct area is 60 (PG,rules I l2-ll4). An ancient commentator of the PG everr goes further and points out an interestingtheoretical defect ofthe rule (l), or (t) other than its grossness. FIe says (P. 160) that therule may yield a rough anlrwerfor thearea even in the caseof irnpossiblefigures, and givesthe example of a triangle of base 20 and flank sides 13 and 7. Since the sum of the twosides is equal to the third (base), no tri:rngle is possible, but the formula (a) will give 100for its gross area. The MS, XV, 69 (p.165) gives the BSS rrrle for the accurate area, but it is laid downth e re for a t r ianglc o rrl y a n d n o t fo r a q u a d ri l ateral . B hi skara II (A .D . l l 50) i n hi sLilivati (eitmadl), rule 169, hasalso given the same rule but with the remark tltat it givesevact area for a triangle and inexact (asphuta) for a quadrilateralG. 3. Sorne Historical and Other Rernarks: The approximate fornrula (l) was used outside India much before thedate of Brahm-agupta. The Babylonians of thc ancient.Mesopotamianvalley are stated to have used itinfinding the area of e quadrilateralT. The seme formula can be gathernd from the inscripti-ons (about 100 B.C.) found on the Iemple of Horus at Edfn8. In this type of Egyptianme n sur at ional m at h e n ra ti c s , tl re tri a n g l e s w e re regardede as cases of quadri l ateral s i nwhich one sido (the face) is mad.: zero, just as what is met with in Brahnragupta. The Chinese mathematical work Wu-tsao Suan-ching( about 5th ot 6rh century )applies the formula (l) for coniputing the area of a quadrangr,,ar field whose eastern,western, southern, and northern sidesare given to be 35, 45r 25, and l5 pac6srespectivelylo. The formula (5) Ibr the area of a triangle is generally ca"lledHerons Formula, but,according to same medival Arattic scholars, it was kuown everr to Archimedes (third centuryB.C .;tt . How Brahmagupta arrived at his formula (2), is difficult to say with certainty. Foran expostion of the attempted prool, of this formula, as given by GaneSa Daivajfra (qivrrien)in his commentary (1545 A.D. l on tle Lilavati, a paper by M. G. Inamdar may be consul-tedlr. The Ytrkti-bhnsl 1-y6, sixteenth century) also contains a proof of the sameformulal3. 4. Brahrnaguptas Expressions for the Diagonals: The BSS, XIl,24 (Vol. III, p. 836) states +<rtFragqqrti+agvaurfr;qqTfqilTqrtq r alrrngwfagsratrql:nqit ca ftq} lrictl Karnl ti rita-bhujaghfitaikyam-rrbhay:rthzin1onya-bhjitam gunayet / logena bhujapratiblrda-vacllravoh krrnau pade visame ll2Sl I The sums of the products of the sidesabout the diagonals be both divided by eachother; multiply (the quotients obtained) by the sum of the prodr cts of the opporite sides;
  4. 4. ED U OAT IO N35 r IIT M ATHEM ATICE 11.the lquare-root (of the results) are the two diagonalr (visame Ihat is AC:,/m (7) BD:/M (B) Brahmaguptas Sanskrit stanza, giving these-diagonals, by has been quote-dra.vcrbatinBhrs126 It in"hii Lilivati witn the remark that although indeternrinate,the diagonals aresought to es determinate by Brahmagupta and others. It may be noted that the, from (7) and (B), we immediately get ...(7) A C . BD :a .c Ib .d ...(8) which is called the Ptolemys Theorem for cyclic quadrilateral after the famous Greek astro- nomer of the secondcentuiy A. D, The YB (pp, 232-33), horlever, frrllcrvs the opposite procedure of deriving (7) and (8) from (9) and someother relations. tmost remarka Brahmagupta.1 expressionsfor the diagonals are considered to be the ble in Hindu gu"o-etry and solitary in its excellence" by a recent historian of mathematicslt The formnla [e) ir stitea to be rediscoveredro Europe by W. Snell (about l6l9 A.Dr). in Refercncee I MS edited by S. Dvivedi, Fasciculur II, p. 165;Benares,19l0 (Braj Bhusan Das & Co.). 2 For a short decription of Brahmaguptas works, see R.C. Gupta, "Brahmagupt_asRule for the Volume of Frustrrm like Solids", The Matl;enaticsEducation,Vol. VI., No4 (D ec em ber 197 2 (,Sc e . B,p .l 1 7 . 3 BSS, XIl, 26-27.Edited by R.S. Sharma and his team, Vol. III, pp B33-34; New Delhi, 1966 (Indian Inst. of Astronomical:rndSanskrit Research). All page refercncesto BSS are according to this edition. 4 PG, rule ll2a. Editecl. with an ancient commentary, by K.S. Shukla, p. 156 of the text; Lucknorv, iS5) (Lutknow IJuiversity). The editor has placed the author bsllvssn ti50 and 950 A.D., while sev.ral earlier scholarsplaced Sridhera before 850 A.D. (see lG introduction, p.xxxviii).5 Ibid., translation, pp. 87-88.6 The Lilivati, part II, p. 156.Edited bv D.V. Apte, Poona, 1937(Anandasrama Sanskrit S er iesNo. 107) .7 C.B. Boyer, A Hi;tory of Mathematics,p.42; New York, l968 (John Wiley).B T.L. Heath, A ManualdCreek Mathematics, 77; Ncw York 1963 (Dover reprint). p.I Ibid., p. 7B.l0 Y. Mikami, The Deuelopment MathcmaticsinCl.ina and Japan, p. 38; Nerv York, 196l of (Chelseareprint).l l Boy er , op. c it . , p . 1 4 9 .12 Nagpur Uaiuersitjt Journal, 1946, No.l I pp. 36-42.13 YB (in Maiayalarn) part I, pp. 247-257. Edited by Rama Varma lVlaru Thampuran and A.R. Aktrileswar Aiyar, Trichur, lg48 (Mangalodayarn Press).l4 Lilsvati, part II, p. lB0.l5 lroward Eves, An Introduction thc History of Mathematics,p.l}7; New York, 1969 (Holt, to Rinchart and Winston),l6 D E. Smith, Historytof Mathemahcs vol. II, p.287; New York, l95B (Dover reprint).